Relationship between a Lie group such as So(3) and its Lie algebra

[MichaelAlexDavM]

TL;DR

I am trying to understand the relation (significance) between the Lie groups such as so(3) and its Lie algebra?

I am just starting a QM course. I hope these are reasonable questions. I have been given my first assignment. I can answer the questions so far but I do not really understand what's going on. These questions are all about so(N) groups, Pauli matrices, Lie brackets, generators and their Lie algebra.

I am very confused about Lie groups and Lie algebras.
1. What is the difference (or connection) between them?
2. How do you get from a Lie group to a Lie algebra?
3. With regard to particle physics, what's so good about Lie algebra anyway?

 

[strangerep]

You're essentially asking us to give you a crash course of Lie groups. You need to start with a good textbook and ask more specific questions as you progress through it. What textbook(s) have been prescribed for your course?

When I was first learning about Lie groups/algebras in relation to physics, I found this book useful:

Greiner & Muller. "Quantum Mechanics: Symmetries".
ISBN 978-3-642-57976-9

 

[vanhees71]

Roughly speaking a Lie group is a group whose group elements depend on some parameters such that you can make sense of differentiation. In physics you get very far by assuming that Lie groups are given as a subgroup of the general linear group on a finite-dimensional real or complex vector space, i.e., it consists of $\mathbb{R}^{n \times n}$ or $\mathbb{C}^{n \times n}$ matrices that depend on some parameter.

E.g., the rotations are $\mathbb{R}^{3 \times 3}$ matrices with $\hat{R}^{\text{T}}=\hat{R}^{-1}$. It's easy to show that they build a group under usual matrix multiplication. From classical mechanics you should know the Euler angles which parametrize the rotations, and it's clear how to take the derivative of a matrix with respect to these angles (just taking the derivatives of the matrix elements).

Now that you have derivatives you can also look at "infinitesimale" transformations, i.e., you can expand the group elements with respect to the parameters around the unit element. Like with tangents on simple curves you can think of tangents at the unit element of the group, and it's easy to show that the corresponding tangent vectors build a Lie algebra with the commutators of the corresponding infinitesimal transformations as Lie bracket.

 

[martinbn]

Just a comment about notations. The groups are usually denoted by capital letters, so it would be $SO(3)$.

 

[vanhees71]

True, the small-letter pendants denote the Lie algebras, i.e., for $\text{SO}(3)$ it's $\text{so}(3) \simeq \text{su}(2)$.

 

[MichaelAlexDavM]

You're essentially asking us to give you a crash course of Lie groups. You need to start with a good textbook and ask more specific questions as you progress through it.

When I was first learning about Lie groups/algebras in relation to physics, I found this book useful:

Greiner & Muller. "Quantum Mechanics: Symmetries".
ISBN 978-3-642-57976-9

It was a very unfocused question, sorry about that. After a lot of reading online, I think I found what I was looking for.

1. What is the difference (or connection) between them?
My attempt at answering Q_1.
The connection between a Lie group and its Lie algebra is the fact that the Lie algebra can be viewed as the tangent space to the Lie group at the identity. The Lie algebra can be considered as a linearization of the Lie group (near the identity element),
2. How do you get from a Lie group to a Lie algebra?
My attempt at answering Q_2.
There is an exponential map from the tangent space to the Lie group. The exponential map provides the “delinearization,” i.e., it takes us back to the Lie group
3. With regard to particle physics, what's so good about Lie algebra anyway?
My attempt at answering Q_3.
The Lie groups are non-linear and difficult to work with while the lie algebra are linear and easier to work with.

Hopefully my answers are not to far of the mark

What textbook(s) have been prescribed for your course?
We have been prescribed a list of essential reading, there are 4 books on the list. One is by L.D Landau, E.M Lifshitz (1977). Quantum Mechanics. I had a quick look through them in the library but did not loan any out as of yet, which I regret because the college is shut down at the moment.

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[fresh_42]

You could also read:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/
https://www.physicsforums.com/insights/pantheon-derivatives-part-iv/